MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pnt Structured version   Visualization version   Unicode version

Theorem pnt 24452
Description: The Prime Number Theorem: the number of prime numbers less than  x tends asymptotically to  x  /  log (
x ) as  x goes to infinity. This is Metamath 100 proof #5. (Contributed by Mario Carneiro, 1-Jun-2016.)
Assertion
Ref Expression
pnt  |-  ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )  ~~> r  1

Proof of Theorem pnt
Dummy variables  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1re 9642 . . . . . . 7  |-  1  e.  RR
21rexri 9693 . . . . . 6  |-  1  e.  RR*
3 1lt2 10776 . . . . . 6  |-  1  <  2
4 df-ioo 11639 . . . . . . 7  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
5 df-ico 11641 . . . . . . 7  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
6 xrltletr 11454 . . . . . . 7  |-  ( ( 1  e.  RR*  /\  2  e.  RR*  /\  w  e. 
RR* )  ->  (
( 1  <  2  /\  2  <_  w )  ->  1  <  w
) )
74, 5, 6ixxss1 11653 . . . . . 6  |-  ( ( 1  e.  RR*  /\  1  <  2 )  ->  (
2 [,) +oo )  C_  ( 1 (,) +oo ) )
82, 3, 7mp2an 678 . . . . 5  |-  ( 2 [,) +oo )  C_  ( 1 (,) +oo )
9 resmpt 5154 . . . . 5  |-  ( ( 2 [,) +oo )  C_  ( 1 (,) +oo )  ->  ( ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )  |`  ( 2 [,) +oo ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) ) )
108, 9mp1i 13 . . . 4  |-  ( T. 
->  ( ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) )  |`  ( 2 [,) +oo ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) ) )
118sseli 3428 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  ( 1 (,) +oo ) )
12 ioossre 11696 . . . . . . . . . . 11  |-  ( 1 (,) +oo )  C_  RR
1312sseli 3428 . . . . . . . . . 10  |-  ( x  e.  ( 1 (,) +oo )  ->  x  e.  RR )
1411, 13syl 17 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  RR )
15 2re 10679 . . . . . . . . . . 11  |-  2  e.  RR
16 pnfxr 11412 . . . . . . . . . . 11  |- +oo  e.  RR*
17 elico2 11698 . . . . . . . . . . 11  |-  ( ( 2  e.  RR  /\ +oo  e.  RR* )  ->  (
x  e.  ( 2 [,) +oo )  <->  ( x  e.  RR  /\  2  <_  x  /\  x  < +oo ) ) )
1815, 16, 17mp2an 678 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  <->  ( x  e.  RR  /\  2  <_  x  /\  x  < +oo ) )
1918simp2bi 1024 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  2  <_  x )
20 chtrpcl 24102 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
( theta `  x )  e.  RR+ )
2114, 19, 20syl2anc 667 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( theta `  x )  e.  RR+ )
22 0red 9644 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) +oo )  ->  0  e.  RR )
231a1i 11 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) +oo )  ->  1  e.  RR )
24 0lt1 10136 . . . . . . . . . . . 12  |-  0  <  1
2524a1i 11 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) +oo )  ->  0  <  1 )
26 eliooord 11694 . . . . . . . . . . . 12  |-  ( x  e.  ( 1 (,) +oo )  ->  ( 1  <  x  /\  x  < +oo ) )
2726simpld 461 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) +oo )  ->  1  < 
x )
2822, 23, 13, 25, 27lttrd 9796 . . . . . . . . . 10  |-  ( x  e.  ( 1 (,) +oo )  ->  0  < 
x )
2913, 28elrpd 11338 . . . . . . . . 9  |-  ( x  e.  ( 1 (,) +oo )  ->  x  e.  RR+ )
3011, 29syl 17 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  RR+ )
3121, 30rpdivcld 11358 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (
theta `  x )  /  x )  e.  RR+ )
3231adantl 468 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  /  x )  e.  RR+ )
33 ppinncl 24101 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  NN )
3414, 19, 33syl2anc 667 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  e.  NN )
3534nnrpd 11339 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  e.  RR+ )
3613, 27rplogcld 23578 . . . . . . . . . 10  |-  ( x  e.  ( 1 (,) +oo )  ->  ( log `  x )  e.  RR+ )
3711, 36syl 17 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  ( log `  x )  e.  RR+ )
3835, 37rpmulcld 11357 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (π `  x )  x.  ( log `  x ) )  e.  RR+ )
3921, 38rpdivcld 11358 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) )  e.  RR+ )
4039adantl 468 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  e.  RR+ )
4130ssriv 3436 . . . . . . . 8  |-  ( 2 [,) +oo )  C_  RR+
42 resmpt 5154 . . . . . . . 8  |-  ( ( 2 [,) +oo )  C_  RR+  ->  ( ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  |`  ( 2 [,) +oo ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( ( theta `  x )  /  x ) ) )
4341, 42ax-mp 5 . . . . . . 7  |-  ( ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  |`  ( 2 [,) +oo ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( ( theta `  x )  /  x ) )
44 pnt2 24451 . . . . . . . 8  |-  ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  ~~> r  1
45 rlimres 13622 . . . . . . . 8  |-  ( ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  ~~> r  1  ->  ( ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  |`  ( 2 [,) +oo ) )  ~~> r  1 )
4644, 45mp1i 13 . . . . . . 7  |-  ( T. 
->  ( ( x  e.  RR+  |->  ( ( theta `  x )  /  x
) )  |`  (
2 [,) +oo )
)  ~~> r  1 )
4743, 46syl5eqbrr 4437 . . . . . 6  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( ( theta `  x
)  /  x ) )  ~~> r  1 )
48 chtppilim 24313 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  |->  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) ) )  ~~> r  1
4948a1i 11 . . . . . 6  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( ( theta `  x
)  /  ( (π `  x )  x.  ( log `  x ) ) ) )  ~~> r  1 )
50 ax-1ne0 9608 . . . . . . 7  |-  1  =/=  0
5150a1i 11 . . . . . 6  |-  ( T. 
->  1  =/=  0
)
5239rpne0d 11346 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) )  =/=  0 )
5352adantl 468 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  =/=  0 )
5432, 40, 47, 49, 51, 53rlimdiv 13709 . . . . 5  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( ( ( theta `  x )  /  x
)  /  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) ) ) )  ~~> r  ( 1  /  1 ) )
5514recnd 9669 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  CC )
56 chtcl 24036 . . . . . . . . . . . . 13  |-  ( x  e.  RR  ->  ( theta `  x )  e.  RR )
5713, 56syl 17 . . . . . . . . . . . 12  |-  ( x  e.  ( 1 (,) +oo )  ->  ( theta `  x )  e.  RR )
5857recnd 9669 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) +oo )  ->  ( theta `  x )  e.  CC )
5911, 58syl 17 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  ( theta `  x )  e.  CC )
6055, 59mulcomd 9664 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  ( x  x.  ( theta `  x
) )  =  ( ( theta `  x )  x.  x ) )
6160oveq2d 6306 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( ( theta `  x )  x.  ( (π `  x )  x.  ( log `  x
) ) )  / 
( x  x.  ( theta `  x ) ) )  =  ( ( ( theta `  x )  x.  ( (π `  x )  x.  ( log `  x
) ) )  / 
( ( theta `  x
)  x.  x ) ) )
6238rpcnd 11343 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (π `  x )  x.  ( log `  x ) )  e.  CC )
6330rpne0d 11346 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  x  =/=  0 )
6421rpne0d 11346 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  ( theta `  x )  =/=  0
)
6562, 55, 59, 63, 64divcan5d 10409 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( ( theta `  x )  x.  ( (π `  x )  x.  ( log `  x
) ) )  / 
( ( theta `  x
)  x.  x ) )  =  ( ( (π `  x )  x.  ( log `  x
) )  /  x
) )
6661, 65eqtrd 2485 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( ( theta `  x )  x.  ( (π `  x )  x.  ( log `  x
) ) )  / 
( x  x.  ( theta `  x ) ) )  =  ( ( (π `  x )  x.  ( log `  x
) )  /  x
) )
6738rpne0d 11346 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (π `  x )  x.  ( log `  x ) )  =/=  0 )
6859, 55, 59, 62, 63, 67, 64divdivdivd 10430 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( ( theta `  x )  /  x )  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) )  =  ( ( ( theta `  x )  x.  ( (π `  x )  x.  ( log `  x
) ) )  / 
( x  x.  ( theta `  x ) ) ) )
6934nncnd 10625 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  e.  CC )
7037rpcnd 11343 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( log `  x )  e.  CC )
7137rpne0d 11346 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( log `  x )  =/=  0
)
7269, 55, 70, 63, 71divdiv2d 10415 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (π `  x )  /  (
x  /  ( log `  x ) ) )  =  ( ( (π `  x )  x.  ( log `  x ) )  /  x ) )
7366, 68, 723eqtr4d 2495 . . . . . 6  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( ( theta `  x )  /  x )  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) )  =  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )
7473mpteq2ia 4485 . . . . 5  |-  ( x  e.  ( 2 [,) +oo )  |->  ( ( ( theta `  x )  /  x )  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )
75 1div1e1 10300 . . . . 5  |-  ( 1  /  1 )  =  1
7654, 74, 753brtr3g 4434 . . . 4  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )  ~~> r  1 )
7710, 76eqbrtrd 4423 . . 3  |-  ( T. 
->  ( ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) )  |`  ( 2 [,) +oo ) )  ~~> r  1 )
78 ppicl 24058 . . . . . . . . . 10  |-  ( x  e.  RR  ->  (π `  x )  e.  NN0 )
7913, 78syl 17 . . . . . . . . 9  |-  ( x  e.  ( 1 (,) +oo )  ->  (π `  x
)  e.  NN0 )
8079nn0red 10926 . . . . . . . 8  |-  ( x  e.  ( 1 (,) +oo )  ->  (π `  x
)  e.  RR )
8129, 36rpdivcld 11358 . . . . . . . 8  |-  ( x  e.  ( 1 (,) +oo )  ->  ( x  /  ( log `  x
) )  e.  RR+ )
8280, 81rerpdivcld 11369 . . . . . . 7  |-  ( x  e.  ( 1 (,) +oo )  ->  ( (π `  x )  /  (
x  /  ( log `  x ) ) )  e.  RR )
8382recnd 9669 . . . . . 6  |-  ( x  e.  ( 1 (,) +oo )  ->  ( (π `  x )  /  (
x  /  ( log `  x ) ) )  e.  CC )
8483adantl 468 . . . . 5  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  (
(π `  x )  / 
( x  /  ( log `  x ) ) )  e.  CC )
85 eqid 2451 . . . . 5  |-  ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )  =  ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )
8684, 85fmptd 6046 . . . 4  |-  ( T. 
->  ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) ) : ( 1 (,) +oo ) --> CC )
8712a1i 11 . . . 4  |-  ( T. 
->  ( 1 (,) +oo )  C_  RR )
8815a1i 11 . . . 4  |-  ( T. 
->  2  e.  RR )
8986, 87, 88rlimresb 13629 . . 3  |-  ( T. 
->  ( ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) )  ~~> r  1  <->  ( (
x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )  |`  (
2 [,) +oo )
)  ~~> r  1 ) )
9077, 89mpbird 236 . 2  |-  ( T. 
->  ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )  ~~> r  1 )
9190trud 1453 1  |-  ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )  ~~> r  1
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    /\ w3a 985    = wceq 1444   T. wtru 1445    e. wcel 1887    =/= wne 2622    C_ wss 3404   class class class wbr 4402    |-> cmpt 4461    |` cres 4836   ` cfv 5582  (class class class)co 6290   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540    x. cmul 9544   +oocpnf 9672   RR*cxr 9674    < clt 9675    <_ cle 9676    / cdiv 10269   NNcn 10609   2c2 10659   NN0cn0 10869   RR+crp 11302   (,)cioo 11635   [,)cico 11637    ~~> r crli 13549   logclog 23504   thetaccht 24017  πcppi 24020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-disj 4374  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-fi 7925  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-fac 12460  df-bc 12488  df-hash 12516  df-shft 13130  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-limsup 13526  df-clim 13552  df-rlim 13553  df-o1 13554  df-lo1 13555  df-sum 13753  df-ef 14121  df-e 14122  df-sin 14123  df-cos 14124  df-pi 14126  df-dvds 14306  df-gcd 14469  df-prm 14623  df-pc 14787  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-pt 15343  df-prds 15346  df-xrs 15400  df-qtop 15406  df-imas 15407  df-xps 15410  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-mulg 16676  df-cntz 16971  df-cmn 17432  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-fbas 18967  df-fg 18968  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cld 20034  df-ntr 20035  df-cls 20036  df-nei 20114  df-lp 20152  df-perf 20153  df-cn 20243  df-cnp 20244  df-haus 20331  df-cmp 20402  df-tx 20577  df-hmeo 20770  df-fil 20861  df-fm 20953  df-flim 20954  df-flf 20955  df-xms 21335  df-ms 21336  df-tms 21337  df-cncf 21910  df-limc 22821  df-dv 22822  df-log 23506  df-cxp 23507  df-em 23918  df-cht 24023  df-vma 24024  df-chp 24025  df-ppi 24026  df-mu 24027
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator